Motion in one dimension

2.1Displacement and velocity

2-1 Objectives1. Describe motion in terms of displacement, time, and

velocity.2. Calculate the displacement of an object traveling at a known

velocity for a specific time interval.3. Construct and interpret graphs of position versus time.

Do Now - Notes Key terms and

ideasnotes

Frame of reference

Displacement

distance

Vector and Scalar

Average velocity

Velocity (instantaneous velocity)

Velocity in a d. vs. t graph

Question

• How far do you travel to get to school in the morning?

• How do you compare this distance to the approximate straight-line distance between their home and school?

• There could be many distances between xf and xi many be many, distance depends on the path.

• There is only one displacement between xf and xi. displacement refers to shortest distance between the xf and xi and direction from xi to xf

Distance vs. Displacement

Displacement = change in position = final position – initial position

∆x = xf - xi

∆ denotes change

xi xf

Scalar vs. Vector• SCALAR

– A measured quantity that has NO DIRECTION– Examples

• Distance, Time, Mass, Volume

• VECTOR– A measured quantity that includes DIRECTION– SIGN SHOWS DIRECTION– Example

• Displacement

Example• A man drives his car 3 miles north, then 4

miles east.

What distance did he travel?What is his displacement from his point of origin?

3 miNorth

4 miEast

Displacement5 mi

Somewhat Northeast

Distance7 mi

Example• Three men leave the same house on foot. The first man walks

30 feet north, then 40 feet west. The second man walks 90 feet south, then 88 feet north. The third man walks 10 feet east, then 50 feet west.

• Which man has traveled the greatest distance?

• Who is farthest from the house?

• Who is closest to the house?

The second man

The first man

The second man

The frame of reference

• http://www.physics-chemistry-interactive-flash-animation.com/mechanics_forces_gravitation_energy_interactive/frame_of_reference_motion_child_ball_train.htm

• The choice of a reference point for the coordinate system is arbitrary, but once chose, the same point must be used throughout the problem.

• Text book, p41, figure 2-2, what would the displacement of the gecko be if the zero end of the meter stick had been lined up with the gecko’s first position?

Positive and negative displacement

How do you describe the speed of the car?The speed changes

Average speedInstantaneous speed

Average Velocity vs. Average Speed

• AVERAGE VELOCITY – change in DISPLACEMENT occurring over time– Includes both MAGNITUDE and DIRECTION

• VECTOR

• The direction of the velocity vector is simply the same as the direction that an object is moving.

• AVERAGE SPEED – change in DISTANCE occurring over time– Includes ONLY MAGNITUDE

• SCALAR

Calculate Average Speed and Average Velocity

• The average speed during the course of a motion is often computed using the following formula:

• In contrast, the average velocity is often computed using this formula

Does NOT include DIRECTION!

if

ifavg tt

xx

t

xvv

The language of physics: ∆ means change

Example• Sally gets up one morning and decides to

take a three mile walk. She completes the first mile in 8.3 minutes, the second mile in 8.9 minutes, and the third mile in 9.2 minutes.– What is her average speed during her walk?

vavg = d / tvavg = 3 mi / (8.3 min + 8.9 min + 9.2 min)

vavg = 0.11 mi / min

Example• Tom gets on his bike at 12:00 pm and

begins riding west. At 12:30 pm he has ridden 8 miles.

– What was his average velocity during his ride?

vavg = d / tvavg = 8 mi / 30 min

vavg = 0.27 mi / min WEST

Example

• During a race on level ground, Andra runs with an average velocity of 6.02 m/s to the east. What displacement does Andre cover in 137 s?

Answer: 825 m East

vavg = ∆x∆t (∆t )(∆t )

∆x = vavg (∆t ) = (6.02 m/s)(137 s) = 825 m

Class work

Practice p. 44 #1-6

Interpret velocity in p-t graph

What does this remind you of?Position vs. Time

Time

Po

sit

ion

What is happening in thisgraph?

SLOPE OF A GRAPH!

CONSTANTZERO

SLOPE

MotionlessObject

Position vs. Time

Time

Po

sit

ion

CONSTANTPOSITIVE

SLOPE

Moving withCONSTANT

positive velocity

Position vs. Time

Time

Po

sit

ion

INCREASINGSLOPE Moving with

INCREASINGvelocity

if

ifavg tt

xx

t

xvv

Graph interpretation of velocity

Alert:Only use points on the line to calculate slope.

Average velocity during 0-55 s

Average velocity during 11-33 s

Distance vs. Time GraphsDuring what time interval was the object NOT MOVING?

Distance vs. Time

0

2

4

6

8

10

12

14

16

18

0 1 2 3 4 5 6 7

Time (s)

Dis

tan

ce (

m)

2 – 3 secondsThe interval on the graph wherethe distance remains constant!

Displacement vs. Time Graphs

Displacement vs. Time

-4

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7

Time (s)

Dis

pla

cem

en

t (m

)

When the displacement isnegative, the object has a

position to the left of the origin1 – 2 and 4 – 5 seconds

Constant displacement means thatthe object doesn’t move

The object’s final position isat +1 meter (1 meter to the

right of the origin)

During what time interval(s) was the object to the left of the origin?During what time interval(s) was the object NOT MOVING?At what distance from the origin does the object stop?

As the slope goes, so goes the velocity

Slow, Positive, Constant Velocity Fast, Positive, Constant Velocity

Slow, Negative Constant Velocity

Fast, Negative, Constant Velocity

example

Determine average velocity

1. during 0-5 seconds

2. During 5-10 seconds

The velocity is 5 m/s between 0-5 seconds

The velocity is zero between 5-10 seconds

Distance-time graph

• NEVER decreases • Read graph to find current

total distance • Subtract points to find

distance traveled between them

• Average speed = slope or Δd/Δt

• curve = changing speed (acceleration or deceleration) – increasing slope =

increasing speed – decreasing slope =

decreasing speed

Displacement-time graph

Above x-axis = positive displacement from origin (east, right, up); Below x-axis = negative displacement from origin (west, left, down)

Read graph to find current position. Difference between points = displacement traveled (change in position); Accumulate to get total distance

velocity = slope or Δd/Δt positive slope = headed in positive direction from origin (east, right, up)negative slope = headed in negative direction from origin (west, left, down)

• curve = changing speed (acceleration or deceleration)

increasing slope = increasing speed decreasing slope = decreasing speed

Distance vs. time graph

Displacement vs. time graph

Speed (Instantaneous Speed) and velocity (Instaneous Velocity)

• Speed, often means instantaneous speed - the speed at any given instant in time. It is often ref

• Velocity (Instantaneous velocity) - the speed at any given instant in time with direction at that instant

• The magnitude of Instantaneous velocity is always equal to the instantaneous speed

• The magnitude of average velocity can be less than the average speed.

Instantaneous velocity

• Text book p. 46 – figure 2-7

• The instantaneous velocity at a given time can be determined by measuring the slope of the line that is tangent to that point on the position versus time graph.

Average speed vs. instantaneous speed on p-t graph

Position vs. Time

Time

Po

sit

ion

The slope of the secant line is between points A and B is the Average velocity between A & B

A

B

The slope of the tangent line is at point A is the instantaneous velocity at point A

CLASS WORK

• Section Review Worksheet 2-1, “Displacement and Velocity.” Graph Skills